Numerical analysis of modular regularization methods for the BDF2 time discretization of the Navier-Stokes equations

Layton, William; Mays, Nathaniel; Neda, Monika; Trenchea, Catalin

ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 3, pp. 765-793.

We consider an uncoupled, modular regularization algorithm for approximation of the Navier-Stokes equations. The method is: Step 1.1: Advance the NSE one time step, Step 1.1: Regularize to obtain the approximation at the new time level. Previous analysis of this approach has been for specific time stepping methods in Step 1.1 and simple stabilizations in Step 1.1. In this report we extend the mathematical support for uncoupled, modular stabilization to (i) the more complex and better performing BDF2 time discretization in Step 1.1, and (ii) more general (linear or nonlinear) regularization operators in Step 1.1. We give a complete stability analysis, derive conditions on the Step 1.1 regularization operator for which the combination has good stabilization effects, characterize the numerical dissipation induced by Step 1.1, prove an asymptotic error estimate incorporating the numerical error of the method used in Step 1.1 and the regularizations consistency error in Step 1.1 and provide numerical tests.

[1] N. Adams and A. Leonard, Deconvolution of subgrid scales for the simulation of shock-turbulence interaction, in Direct and Large Eddys Simulation III, edited by N.S.P. Voke and L. Kleiser. Kluwer, Dordrecht (1999) 201.

[2] N. Adams and S. Stolz, Deconvolution methods for subgrid-scale approximation in large-eddy simulation, Modern Simulation Strategies for Turbulent Flow, edited by R.T. Edwards (2001).

[3] N. Adams and S. Stolz, A subgrid-scale deconvolution approach for shock capturing. J. Comput. Phys. 178 (2002) 391-426. | MR | Zbl

[4] G.A. Baker, V.A. Dougalis and O.A. Karakashian, On a higher order accurate fully discrete Galerkin approximation to the Navier-Stokes equations. Math. Comput. 39 (1982) 339-375. | MR | Zbl

[5] D. Barbato, L.C. Berselli and C.R. Grisanti, Analytical and numerical results for the rational large eddy simulation model. J. Math. Fluid Mech. 9 (2007) 44-74. | MR | Zbl

[6] L.C. Berselli, On the large eddy simulation of the Taylor-Green vortex. J. Math. Fluid Mech. 7 (2005) S164-S191. | MR | Zbl

[7] J.P. Boyd, Two comments on filtering (artificial viscosity) for Chebyshev and Legendre spectral and spectral element methods: preserving boundary conditions and interpretation of the filter as a diffusion. J. Comput. Phys. 143 (1998) 283-288. | MR | Zbl

[8] S.C. Brenner and L.R. Scott, The mathematical theory of finite element methods, vol. 15 of Texts in Applied Mathematics. Springer-Verlag, New York (1994). | MR | Zbl

[9] C. Canuto, M.Y. Hussaini, A. Quarteroni and T.A. Zang, Spectral methods, Evolution to complex geometries and applications to fluid dynamics. Scientific Computation. Springer, Berlin (2007). | MR | Zbl

[10] A. Chorin, Numerical solution for the Navier-Stokes equations. Math. Comput. 22 (1968) 745-762. | Zbl

[11] J. Connors and W. Layton, On the accuracy of the finite element method plus time relaxation. Math. Comput. 79 (2010) 619-648. | MR | Zbl

[12] A. Dunca, Investigation of a shape optimization algorithm for turbulent flows, tech. rep., Argonne National Lab, report number ANL/MCS-P1101-1003 (2002). Available at http://www-fp.mcs.anl.gov/division/publications/.

[13] A. Dunca, Space averaged Navier Stokes equations in the presence of walls. Ph.D. thesis, University of Pittsburgh (2004). | MR

[14] A. Dunca and Y. Epshteyn, On the Stolz-Adams deconvolution model for the large-eddy simulation of turbulent flows. SIAM J. Math. Anal. 37 (2006) 1890-1902. | MR | Zbl

[15] E. Emmrich, Error of the two-step BDF for the incompressible Navier-Stokes problem. M2AN: M2AN 38 (2004) 757-764. | Numdam | MR | Zbl

[16] V. Ervin, W. Layton and M. Neda, Numerical analysis of a higher order time relaxation model of fluids. Int. J. Numer. Anal. Model. 4 (2007) 648-670. | MR | Zbl

[17] V. Ervin, W. Layton and M. Neda, Numerical analysis of filter based stabilization for evolution equations. SINUM 50 (2012) 2307-2335. | MR | Zbl

[18] P. Fischer and J. Mullen, Filter-based stabilization of spectral element methods. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001) 265-270. | MR | Zbl

[19] E. Garnier, N. Adams and P. Sagaut, Large eddy simulation for compressible flows. Sci. Comput. Springer, Berlin (2009). | MR | Zbl

[20] V. Girault and P.-A. Raviart, Finite element approximation of the Navier-Stokes equations, in vol. 749 of Lect. Notes Math. Springer-Verlag, Berlin (1979). | MR | Zbl

[21] M.D. Gunzburger, Finite element methods for viscous incompressible flows, A guide to theory, practice, and algorithms. Computer Science and Scientific Computing. Academic Press Inc., Boston, MA (1989). | MR | Zbl

[22] F. Hecht and O. Pironneau, Freefem++, webpage: http://www.freefem.org.

[23] J.G. Heywood and R. Rannacher, Finite-element approximation of the nonstationary Navier-Stokes problem. IV. Error analysis for second-order time discretization. SIAM J. Numer. Anal. 27 (1990) 353-384. | MR | Zbl

[24] V. John, Large eddy simulation of turbulent incompressible flows, Analytical and numerical results for a class of LES models, in vol. 34 of Lect. Notes Comput. Sci. Engrg. Springer-Verlag, Berlin (2004). | MR | Zbl

[25] V. John and W.J. Layton, Analysis of numerical errors in large eddy simulation. SIAM J. Numer. Anal. 40 (2002) 995-1020. | MR | Zbl

[26] W. Layton, Superconvergence of finite element discretization of time relaxation models of advection. BIT 47 (2007) 565-576. | MR | Zbl

[27] W. Layton, The interior error of van Cittert deconvolution is optimal. Appl. Math. 12 (2012) 88-93. | MR | Zbl

[28] W. Layton, C. Manica, M. Neda and L. Rebholz, Helicity and energy conservation and dissipation in approximate deconvolution LES models of turbulence. Adv. Appl. Fluid Mech. 4 (2008) 1-46. | MR | Zbl

[29] W. Layton and M. Neda, Truncation of scales by time relaxation. J. Math. Anal. Appl. 325 (2007) 788-807. | MR | Zbl

[30] W. Layton, L.G. Rebholz and C. Trenchea, Modular nonlinear filter stabilization of methods for higher Reynolds numbers flow. J. Math. Fluid Mech. (2011) 1-30. | MR | Zbl

[31] W. Layton, L. Röhe and H. Tran, Explicitly uncoupled VMS stabilization of fluid flow. Comput. Methods Appl. Mech. Engrg. 200 (2011) 3183-3199. | MR | Zbl

[32] J. Mathew, R. Lechner, H. Foysi, J. Sesterhenn and R. Friedrich, An explicit filtering method for large eddy simulation of compressible flows. Phys. Fluids 15 (2003). | Zbl

[33] J.S. Mullen and P.F. Fischer, Filtering techniques for complex geometry fluid flows. Commun. Numer. Methods Engrg. 15 (1999) 9-18. | MR | Zbl

[34] S. Ravindran, Convergence of extrapolated BDF2 finite element schemes for unsteady penetrative convection model. Numer. Funct. Anal. Optim. 33 (2011) 48-79. | MR | Zbl

[35] P. Rosenau, Extending hydrodynamics via the regularization of the Chapman-Enskog expansion. Phys. Rev. A 40 (1989) 7193. | MR

[36] M. Schäfer and S. Turek, Benchmark computations of laminar flow around cylinder, in Flow Simulation with High-Performance Computers II, vol. 52. Edited by H. EH. Vieweg (1996) 547-566. | Zbl

[37] S. Schochet and E. Tadmor, The regularized Chapman-Enskog expansion for scalar conservation laws. Arch. Rat. Mech. Anal. 119 (1992) 95. | MR | Zbl

[38] I. Stanculescu, Existence theory of abstract approximate deconvolution models of turbulence. Ann. Univ. Ferrara Sez. VII Sci. Mat. 54 (2008) 145-168. | MR | Zbl

[39] S. Stolz and N. Adams, On the approximate deconvolution procedure for LES. Phys. Fluids, II (1999) 1699-1701. | Zbl

[40] S. Stolz, N. Adams and L. Kleiser, An approximate deconvolution model for large eddy simulation with application to wall-bounded flows. Phys. Fluids 13 (2001) 997-1015. | Zbl

[41] S. Stolz, N. Adams and L. Kleiser, The approximate deconvolution model for LES of compressible flows and its application to shock-turbulent-boundary-layer interaction. Phys. Fluids 13 (2001) 2985. | Zbl

[42] S. Stolz, N. Adams and L. Kleiser, The approximate deconvolution model for compressible flows: isotropic turbulence and shock-boundary-layer interaction,Advances in LES of Complex Flows, in vol. 65 of Fluid Mechanics and Its Applications. Edited by R. Friedrich and W. Rodi. Springer, Netherlands (2002) 33-47. | Zbl

[43] D. Tafti, Comparison of some upwind-biased high-order formulations with a second-order central-difference scheme for time integration of the incompressible Navier-Stokes equations. Comput. Fluids 25 (1996) 647-665. | MR | Zbl

[44] G. Taylor, On decay of vortices in a viscous fluid. Phil. Mag. 46 (1923) 671-674. | JFM

[45] G.I. Taylor and A.E. Green, Mechanism of the production of small eddies from large ones, Proc. Royal Soc. London Ser. A 158 (1937) 499-521. | JFM

[46] M. Visbal and D. Rizzetta, Large-eddy simulation on general geometries using compact differencing and filtering schemes, AIAA Paper (2002) 2002-288.

[47] X. Wang, An efficient second order in time scheme for approximating long time statistical properties of the two dimensional Navier-Stokes equations. Numer. Math. 121 (2012) 753-779. | MR

[48] E. Zeidler, Applied functional analysis, vol. 108 of Appl. Math. Sci. Springer-Verlag, New York (1995). | MR | Zbl